What is the integral of (e^x)/(1-sqrt(e^x)) ?

The integral of (e^x)/(1-sqrt(e^x)) is -2(-1+e^{x/2}+ln|1-e^{x/2}|)+C

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integral of (e^x)/(1-sqrt(e^x))

\int \frac{e ^{x}}{1 - e ^{x}} d x

- 2 (- 1 + e^{\frac{x}{2}} + ln 1 - e^{\frac{x}{2}}) + C

Solution steps

\int \frac{e ^{x}}{1 - e ^{x}} d x

Apply u-substitution

: \int \frac{1}{1 - u} d u

= \int \frac{1}{1 - u} d u

Apply u-substitution

: \int - \frac{2 ( - v + 1 )}{v} d v

= \int - \frac{2 ( - v + 1 )}{v} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 \cdot \int \frac{- v + 1}{v} d v

Expand

\frac{- v + 1}{v} : - 1 + \frac{1}{v}

= - 2 \cdot \int - 1 + \frac{1}{v} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 2 (- \int 1 d v + \int \frac{1}{v} d v)

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= - 2 (- v + ln ∣ v ∣)

Substitute back

= - 2 (- (1 - e^{x}) + ln 1 - e^{x})

Simplify

- 2 (- (1 - e^{x}) + ln 1 - e^{x}) : - 2 (- 1 + e^{\frac{x}{2}} + ln 1 - e^{\frac{x}{2}})

= - 2 (- 1 + e^{\frac{x}{2}} + ln 1 - e^{\frac{x}{2}})

Add a constant to the solution

= - 2 (- 1 + e^{\frac{x}{2}} + ln 1 - e^{\frac{x}{2}}) + C

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What is the integral of (e^x)/(1-sqrt(e^x)) ?
The integral of (e^x)/(1-sqrt(e^x)) is -2(-1+e^{x/2}+ln|1-e^{x/2}|)+C